Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a set of equations with multiple unknown variables. In IGCSE mathematics, you are usually asked to find the values of two unknowns that satisfy both linear equations at the same time.

联立方程组是包含多个未知数的一组方程。在IGCSE数学中，通常要求你找出同时满足两个线性方程的两个未知数的值。

1. What Are Simultaneous Equations? | 什么是联立方程组？

When you have two equations such as 2x + y = 7 and x − y = 2, you need to find a single pair (x, y) that makes both equations true. This pair is called the solution of the simultaneous equations.

当你有两个方程，例如 2x + y = 7 和 x − y = 2，你需要找到一对 (x, y) 同时使两个方程成立。这一对值就叫做联立方程组的解。

2x + y = 7 ①
x − y = 2 ②

Notice that equation ② can be rearranged to x = y + 2. Substituting this into ① gives a single equation in y. This is the idea behind the substitution method.

注意方程②可以变形为 x = y + 2。将其代入①就得到一个只含 y 的方程。这就是代入法的基本思想。

2. The Elimination Method | 消元法

The elimination method removes one variable by adding or subtracting the equations. To do this, the coefficients of one variable must be the same (or opposite) in both equations.

消元法通过相加或相减两个方程来消去一个变量。要做到这一点，某个变量的系数在两个方程中必须相同（或互为相反数）。

Example | 例： Solve 3x + 2y = 12 and 5x − 2y = 4.

Step 1: Add the two equations to eliminate y.

3x + 2y + 5x − 2y = 12 + 4
8x = 16
x = 2

Step 2: Substitute x = 2 into one of the original equations.

3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3

So the solution is x = 2, y = 3. Always check by substituting both values into the other equation.

所以解是 x = 2，y = 3。务必把这两个值代入另一个方程进行检验。

Sometimes you must multiply one or both equations first to make the coefficients match. For example, solve 2x + 3y = 8 and 3x + 2y = 7. Here neither coefficient matches, so multiply the first equation by 3 and the second by 2.

有时你需要先将一个或两个方程乘以某个数，使系数一致。例如，解 2x + 3y = 8 和 3x + 2y = 7。这里没有系数相同，所以将第一个方程乘以3，第二个方程乘以2。

6x + 9y = 24
6x + 4y = 14

Now subtract: 5y = 10, so y = 2. Then substitute back to get x = 1.

现在相减：5y = 10，所以 y = 2。然后代回得 x = 1。

3. The Substitution Method | 代入法

The substitution method is useful when one variable has coefficient 1. Rearrange one equation to make that variable the subject, then substitute the expression into the other equation.

当某个变量的系数为1时，代入法非常方便。将其中一个方程变形，使该变量成为主体，然后将表达式代入另一个方程。

Example | 例： Solve y = 2x − 1 and 3x + 2y = 12.

Step 1: Substitute y = 2x − 1 into 3x + 2y = 12.

3x + 2(2x − 1) = 12
3x + 4x − 2 = 12
7x = 14
x = 2

Step 2: Substitute x = 2 back into y = 2x − 1 to get y = 3.

第二步：将 x = 2 代回 y = 2x − 1，得到 y = 3。

The substitution method is also essential when solving a linear equation and a quadratic equation together, which appears in the extended IGCSE syllabus.

在IGCSE扩展课程中，联立方程组可能包含一个线性方程和一个二次方程，此时必须使用代入法。

y = x² − 2x + 1 与 y = x + 1

By substituting y = x + 1 into the quadratic, you obtain x + 1 = x² − 2x + 1, which simplifies to x² − 3x = 0. Factorise to get x(x − 3) = 0, so x = 0 or x = 3. Then the corresponding y values are 1 and 4.

将 y = x + 1 代入二次方程，得到 x + 1 = x² − 2x + 1，化简为 x² − 3x = 0。因式分解得 x(x − 3) = 0，所以 x = 0 或 x = 3。对应的 y 值分别是 1 和 4。

4. Geometric Meaning of the Solution | 解的几何意义

Each linear equation represents a straight line on the xy-plane. The solution of the simultaneous equations is the point where the two lines intersect.

每个线性方程在 xy 平面上表示一条直线。联立方程组的解就是两条直线的交点。

Case \| 情况	Number of Solutions \| 解的个数	Meaning \| 含义
Lines intersect at one point 两线交于一点	One unique solution 唯一解	Different gradients 斜率不同
Lines are parallel 两线平行	No solution 无解	Same gradient, different intercept 斜率相同，截距不同
Lines coincide 两线重合	Infinitely many solutions 无穷多解	Same gradient and intercept 斜率和截距都相同

For example, the equations y = 2x + 1 and y = 2x − 3 are parallel, so no solution exists. If you try to solve them by elimination, you will end up with a false statement like 0 = 4.

例如，方程 y = 2x + 1 和 y = 2x − 3 是平行的，因此无解。如果你尝试用消元法求解，最终会得到一个错误等式，如 0 = 4。

5. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解和无穷多解

When solving simultaneous equations, you may encounter special outcomes that indicate no solution or infinitely many solutions.

在解联立方程组时，你可能会遇到特殊结果，表示无解或有无穷多解。

No solution | 无解： If elimination leads to a contradiction such as 0 = 5, the lines are parallel and never meet.

无解： 如果消元后出现矛盾，如 0 = 5，则两线平行，永不相交。

Infinite solutions | 无穷多解： If elimination leads to an identity such as 0 = 0, the two equations represent the same line, so every point on the line is a solution.

无穷多解： 如果消元后得到恒等式，如 0 = 0，则两个方程表示同一条直线，直线上每个点都是解。

2x + 4y = 8 与 x + 2y = 4

Multiplying the second equation by 2 gives exactly the first equation. Hence they are the same line, and there are infinitely many solutions.

将第二个方程乘以2后正好得到第一个方程。因此它们是同一条直线，有无穷多解。

6. Solving Word Problems | 应用题解题

Many IGCSE questions present simultaneous equations in a real-world context. The key is to define variables clearly and translate the information into two equations.

许多IGCSE题目以实际情境呈现联立方程组。关键是清晰地定义变量，并将信息转化为两个方程。

Example | 例： The total cost of 3 apples and 2 bananas is $7. The cost of 1 apple and 4 bananas is $9. Find the cost of one apple and one banana.

设一个苹果 x 美元，一个香蕉 y 美元。根据题意：

3x + 2y = 7
x + 4y = 9

Using elimination, multiply the second equation by 3: 3x + 12y = 27. Subtract the first: 10y = 20, so y = 2. Substitute y = 2 into x + 4(2) = 9, giving x = 1. Therefore an apple costs $1 and a banana costs $2.

使用消元法，将第二个方程乘以3：3x + 12y = 27。减去第一个方程：10y = 20，所以 y = 2。将 y = 2 代入 x + 4(2) = 9，得 x = 1。因此一个苹果1美元，一个香蕉2美元。

Always state your answer clearly with the correct units.

务必用正确的单位清晰地写出答案。

7. Common Mistakes and Tips | 常见错误与技巧

Sign errors | 符号错误： When subtracting equations, be careful with negative terms. For example, subtracting 2x − 3y = 5 from 5x + 2y = 8 means subtracting 2x, −3y and 5 separately.

符号错误： 相减方程时，注意负项。例如，从 5x + 2y = 8 中减去 2x − 3y = 5，需要分别减去 2x、−3y 和 5。
Not checking | 不检验： Always substitute your solution into both original equations to confirm.

不检验： 始终将解代入两个原方程进行确认。
Forgetting to multiply all terms | 忘记乘以所有项： When scaling an equation, multiply every term on both sides.

忘记乘以所有项： 对方程进行缩放时，两边每一项都要乘以相同数。
Choosing the wrong method | 选错方法： Use elimination when coefficients are easy to match; use substitution when one variable has coefficient 1.

选错方法： 当系数容易配平时用消元法；当某个变量系数为1时用代入法。

8. Practice Makes Perfect | 熟能生巧

Try to solve the following pairs of equations on your own, then check your answers.

请尝试自己解下列方程组，然后核对答案。

(a) 4x + y = 11, 2x − y = 5
(b) 3x − 2y = 1, x + 2y = 11
(c) y = 3x − 2, y = x + 6

Answers: (a) x = 3, y = −1 (b) x = 3, y = 4 (c) x = 4, y = 10

答案：(a) x = 3，y = −1 (b) x = 3，y = 4 (c) x = 4，y = 10

For more challenging practice, try solving a linear equation together with a quadratic equation using substitution, and sketch the graphs to verify your solutions.

为了更高挑战，尝试用代入法解一个线性方程与一个二次方程组成的方程组，并通过画图来验证你的解。

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Solving Simultaneous Equations | 解联立方程组